Wavelet-based approximation for two-dimensional singularly perturbed elliptic problems

Abstract

In this research article, we explore the application of non-uniform Haar wavelets (NUHWs) to tackle a two-dimensional singularly perturbed elliptic problem (TDSPEP) on an exponentially graded mesh (EGM). The solution to the type of problem considered in this article exhibits boundary layers near the edges x=0 and y=0. Using an EGM, one can resolve the solution’s boundary layer behavior. The proposed method involves the approximation of the highest-order mixed derivative using NUHWs, followed by successive integration to obtain an approximate Haar solution. We conducted an error analysis for the proposed method, which demonstrates that it achieves second-order accuracy. The performance of the method is corroborated in practice through numerical experiments on three examples. The method can be easily used in any programming language.